Simplify (5 square root of 2x+ square root of 5)(-4 square root of 2x+ square root of 5x)

$(5 \sqrt{2 x} + \sqrt{5}) (- 4 \sqrt{2 x} + \sqrt{5 x})$

Step 1

Expand $(5 \sqrt{2 x} + \sqrt{5}) (- 4 \sqrt{2 x} + \sqrt{5 x})$ using the FOIL Method.

Apply the distributive property.

$5 \sqrt{2 x} (- 4 \sqrt{2 x} + \sqrt{5 x}) + \sqrt{5} (- 4 \sqrt{2 x} + \sqrt{5 x})$

Apply the distributive property.

$5 \sqrt{2 x} (- 4 \sqrt{2 x}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x} + \sqrt{5 x})$

Apply the distributive property.

$5 \sqrt{2 x} (- 4 \sqrt{2 x}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Step 2

Simplify each term.

Multiply $5 \sqrt{2 x} (- 4 \sqrt{2 x})$ .

Multiply $- 4$ by $5$ .

$- 20 \sqrt{2 x} \sqrt{2 x} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Raise $\sqrt{2 x}$ to the power of $1$ .

$- 20 ({\sqrt{2 x}}^{1} \sqrt{2 x}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Raise $\sqrt{2 x}$ to the power of $1$ .

$- 20 ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1}) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 20 {\sqrt{2 x}}^{1 + 1} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Add $1$ and $1$ .

$- 20 {\sqrt{2 x}}^{2} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

$- 20 {({(2 x)}^{\frac{1}{2}})}^{2} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 20 {(2 x)}^{\frac{1}{2} \cdot 2} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Combine $\frac{1}{2}$ and $2$ .

$- 20 {(2 x)}^{\frac{2}{2}} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- 20 {(2 x)}^{\frac{2}{2}} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Rewrite the expression.

$- 20 {(2 x)}^{1} + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Simplify.

$- 20 (2 x) + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Multiply $2$ by $- 20$ .

$- 40 x + 5 \sqrt{2 x} \sqrt{5 x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Multiply $5 \sqrt{2 x} \sqrt{5 x}$ .

Combine using the product rule for radicals.

$- 40 x + 5 \sqrt{5 x (2 x)} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Multiply $2$ by $5$ .

$- 40 x + 5 \sqrt{10 x \cdot x} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Raise $x$ to the power of $1$ .

$- 40 x + 5 \sqrt{10 (x^{1} x)} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Raise $x$ to the power of $1$ .

$- 40 x + 5 \sqrt{10 (x^{1} x^{1})} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 40 x + 5 \sqrt{10 x^{1 + 1}} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Add $1$ and $1$ .

$- 40 x + 5 \sqrt{10 x^{2}} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Reorder $10$ and $x^{2}$ .

$- 40 x + 5 \sqrt{x^{2} \cdot 10} + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Pull terms out from under the radical.

$- 40 x + 5 (x \sqrt{10}) + \sqrt{5} (- 4 \sqrt{2 x}) + \sqrt{5} \sqrt{5 x}$

Multiply $\sqrt{5} (- 4 \sqrt{2 x})$ .

Combine using the product rule for radicals.

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{5 (2 x)} + \sqrt{5} \sqrt{5 x}$

Multiply $5$ by $2$ .

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{5} \sqrt{5 x}$

Multiply $\sqrt{5} \sqrt{5 x}$ .

Combine using the product rule for radicals.

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{5 (5 x)}$

Multiply $5$ by $5$ .

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{25 x}$

Rewrite $25$ as $5^{2}$ .

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + \sqrt{5^{2} x}$

Pull terms out from under the radical.

$- 40 x + 5 x \sqrt{10} - 4 \sqrt{10 x} + 5 \sqrt{x}$

Simplify (5 square root of 2x+ square root of 5)(-4 square root of 2x+ square root of 5x)

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